Problem: The following system of equations is represented by the matrix equation $\text{A}\left[\begin{array} {ccc} x \\ y \\ z \end{array} \right]=\vec{b}$. $\begin{aligned}4x+2y&=1 \\3x+7y+4z&=-4 \\8x+2z&=12\end{aligned}$ ${A}=$ $\vec{b}=$ Represent each row and column in the order in which the variables and equations appear.
Answer: The Strategy A system of equations can be represented by a matrix equation $\text{A}\vec{x}=\vec{b}$, where $\text{A}$ is the coefficient matrix, $\vec{x}$ is the variables vector, and $\vec{b}$ is the constants vector. Each row of the matrix equation represents an equation in the system. [I need an explanation, please!] Representing the system of equations as a matrix equation We are given the system of equations: $\begin{aligned}4x+2y&=1 \\3x+7y+4z&=-4 \\8x+2z&=12\end{aligned}$ First, let's rewrite this system to show the coefficients of each variable. $\begin{aligned}{4}x+{2}y+{0}z&=1 \\{3}x+{7}y+{4}z&=-4 \\{8}x+{0}y+{2}z&=12\end{aligned}$ Now, the coefficient matrix can be written as follows. $\left[\begin{array} {ccc} {4} & {2} & {0} \\ {3} & {7} & {4} \\ {8} & {0} & {2} \end{array} \right]$ We can multiply this matrix by a column vector of variables and set it equal to a column vector with the values on the right side of the equations, as follows. $\left[\begin{array} {ccc} {4} & {2} & {0} \\ {3} & {7} & {4} \\ {8} & {0} & {2} \end{array} \right]\left[\begin{array} {ccc} x \\ y \\ z \end{array} \right] =\left[\begin{array} {ccc} 1 \\ -4 \\ 12 \end{array} \right]$ This is our matrix equation. Summary $\text{A}$ and $\vec{b}$ are shown below. $\text{A}=\left[\begin{array} {ccc} 4 & 2 & 0 \\ 3 & 7 & 4 \\ 8 & 0 & 2 \end{array} \right]~~~~~~~~~~~~ \vec{b}=\left[\begin{array} {ccc} 1 \\ -4 \\ 12 \end{array} \right]$